On the relationship of gerbes to the odd families index theorem

规范理论中的类格结构构造

A gerbe-like construction in gauge theory

本文证明同伦K3曲面族中沿纤维切丛与自对调和2形式丛的直和具有典范自旋结构，通过族Seiberg–Witten方程构造O(1)-格提升。

gauge理论II：同调环面的情况

A gerbe-like construction in gauge theory II: the case of homology tori

本文研究同调环面的奇确定性，通过构造反线性Z/4作用，恢复Baraglia(2023)关于闭合spin4流形mod2Seiberg-Witten不变量的部分结果。

Higher-Degree Analogs of the Determinant Line Bundle

In the first part of this paper, given a smooth family of Dirac-type operators on an odd-dimensional closed manifold, we construct an abelian gerbe-with-connection whose curvature is the three-form component of the Atiyah-Singer families index theorem. In the second part of the paper, given a smooth family of Dirac-type operators whose index lies in the i-th filtration of the reduced K-theory of the parametrizing space, we construct a set of Deligne cohomology class of degree i whose curvatures are the i-form component of the Atiyah-Singer families index theorem.

Conformal boundary operators, T-curvatures, and conformal fractional Laplacians of odd order

We construct continuously parametrised families of conformally invariant boundary operators on densities. These may also be viewed as conformally covariant boundary operators on functions and generalise to higher orders the first-order conformal Robin operator and an analogous third-order operator of Chang-Qing. Our families include operators of critical order on odd-dimensional boundaries. Combined with the (conformal Laplacian power) GJMS operators, a suitable selection of the boundary operators yields formally self-adjoint elliptic conformal boundary problems. Working on a conformal manifol…

Index theory, eta forms, and Deligne cohomology

The Chern classes of a K-theory class which is represented by a vector bundle with connection admit refinements to Cheeger-Simons classes in Deligne cohomology. In the present paper we consider similar refinements in the case where the classes in K-theory are represented by geometric families of Dirac operators. In low dimensions these refinements correspond to the exponentiated eta-invariant, the determinant line bundle with Quillen metric and Bismut-Freed connection, and Lott's index gerbe with connection and curving. We give a unified treatement of these cases as well as their higher genera…

球面上的无穷多族Sasaki-Einstein度量

Infinitely many families of Sasaki-Einstein metrics on spheres

本文证明了维数至少为5的奇数维标准球面及界定可平行化流形的奇数维怪球面上均存在无穷多族Sasaki-Einstein度量。

Renormalised Chern-Weil forms associated with families of Dirac operators

We provide local expressions for Chern-Weil type forms built from superconnections associated with families of Dirac operators previously investigated in work by S. Scott and later work by S. Scott and the second author. When the underlying fibration of manifolds is trivial, the even degree forms can be interpreted as renormalised Chern-Weil forms in as far as they coincide with regularised Chern-Weil forms up to residue correction terms. Similarly, a new formula for the curvature of the local fermionic vacuum line bundles is derived using a residue correction term added to the naive curvature…

Eta forms and the odd pseudodifferential families index

Let $A(t)$ be an elliptic, product-type suspended (which is to say parameter-dependant in a symbolic way) family of pseudodifferential operators on the fibres of a fibration $ϕ$ with base $Y.$ The standard example is $A+it$ where $A$ is a family, in the usual sense, of first order, self-adjoint and elliptic pseudodifferential operators and $t\in\bbR$ is the `suspending' parameter. Let $π_{\cA}:\cA(ϕ)\longrightarrow Y$ be the infinite-dimensional bundle with fibre at $y\in Y$ consisting of the Schwartz-smoothing perturbations, $q,$ making $A_y(t)+q(t)$ invertible for all $t\in\bbR.$ The total e…

Equivariant spectral flow and a Lefschetz theorem on odd dimensional spin manifolds

A notion of equivariant spectral flows for families of self-dual elliptic operators on Riemannian manifolds is purposed. As a consequence, a local version of a Lefschetz fix point theorem is proved for Toeplitz operators on odd-dimensional spin manifolds.

Two nontrivial index theorems in odd dimensions

A recent anomaly computation of Horava and Witten is proved and generalized in the form of two index theorems in odd dimensions. Theorem A is a fixed point formula for orientation-reversing involutions. Theorem B is an index theorem for manifolds with boundary using local boundary conditions. Both hold for families of operators.

具有奇数 $b_+$ 的可约 $4$ 流形的奇异微分同胚

Exotic diffeomorphisms of reducible $4$-manifolds with odd $b_+$

利用族Bauer-Furuta不变量，在具有奇数$b_+$的可约$4$流形上构造奇异微分同胚，并发现映射类群非有限生成的单连通$4$流形新例子。

Abelian Gauge Theory, Knots and Odd Khovanov Homology

A homological invariant of 3-manifolds is defined, using abelian Yang-Mills gauge theory. It is shown that the construction, in an appropriate sense, is functorial with respect to the families of 4-dimensional cobordisms. This construction and its functoriality are used to define several link invariants. The strongest version of these invariants has the form of a filtered chain complex that can recover Khovanov homology of the mirror image as a bi-graded group.

Positively Curved Cohomogeneity One Manifolds and 3-Sasakian Geometry

We give an exhaustive description of all simply connected odd dimensional cohomogeneity one manifolds that can possibly support an invariant metric with positive sectional curvature. Among the known examples of odd dimensional manifolds with positive curvature, apart from spheres, there are two infinite families among the 7-dimensional Eschenburg spaces and 13-dimensional Bazaikin spaces and one isolated 7-dimensional Berger space with this property. In addition to these examples, it turns out that only one isolated 7-manifold, and two infinite 7-dimensional families, potentially admit invaria…

Constraints on families of smooth 4-manifolds from Bauer-Furuta invariants

We obtain constraints on the topology of families of smooth $4$-manifolds arising from a finite dimensional approximation of the families Seiberg-Witten monopole map. Amongst other results these constraints include a families generalisation of Donaldson's diagonalisation theorem and Furuta's $10/8$ theorem. As an application we construct examples of continuous $\mathbb{Z}_p$-actions for any odd prime $p$, which can not be realised smoothly. As a second application we show that the inclusion of the group of diffeomorphisms into the group of homeomorphisms is not a weak homotopy equivalence for …

从统计流形构造指数族

Construction of Exponential Families from Statistical Manifolds

本文通过构造性方法，揭示了统计流形与指数族之间的联系，证明非平坦三维叶具有co-Kähler结构，并能作为显式指数族参数化。

Sasakian immersions into the sphere

The aim of this paper is to study Sasakian immersions of compact Sasakian manifolds into the odd-dimensional sphere equipped with the standard Sasakian structure. We obtain a complete classification of such manifolds in the Einstein and $η$-Einstein cases when the codimension of the immersion is $4$. Moreover, we exhibit infinite families of compact Sasakian $η$--Einstein manifolds which cannot admit a Sasakian immersion into any odd-dimensional sphere. Finally, we show that, after possibly performing a ${\mathcal {D}}$-homothetic deformation, a homogeneous Sasakian manifold can be Sasakian im…